1 Word consists of meaning and time.
Refer to the following paper.
2 Now word is expressed by cylinder.
In category, cylinder is made by the following.
X is topological space that has base point x0.
I is unit closed interval. I = {t ｜0≦t≦1}
Cylinder is expressed by X×I.
Here t is time of word.
X × 0 is meaning of starting point. X × 1 is meaning of ending point.
3 Now Suspension ΣX of topological space X is presented.
ΣX is made by shortening of upper surface X × 0 and under surface X × 1 of cylinder X × I.
4 Now Reduced suspension SX is presented.
SX= S1 ×X
S1 is circumference that is stuck together of point 0 and 1 of closed interval I.
Here SSn =Sn+1 Sn is n-dimensional sphere.
SX is made by shortening of x0 × I from ΣX.
SX is expressed by point.
The point is base point of SX.
5 SX defines group in category.
6 SX has operation of group.
7 Now word follows operation of group in algebra.
8 Connection of words generates sentence.
9 Connection is made by the following.
Now H (I, X) is presented.
H ( I, X) is a road space of topological space X.
H ( I, X) is a set that consists of continuous mapping ψ: I → X at interval I = [0, 1].
Road isψ∈ H (I, X).
Ifψisψ(0) =ψ(1),space X becomes loop space ΩX.
10 If a topological space R is presented, H ( X, ΩR) becomes group.
11 If an Abelian group A is presented, space Kn (n = 1,2,3….), a set H(X, Kn) becomes Abelian space.
This space is identified with connection of words, namely, sentence.
Tokyo April 13, 2007
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